reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem Th9:
  i1*i,i2*i are_congruent_mod i3 & i,i3 are_coprime implies
  i1,i2 are_congruent_mod i3
proof
  assume that
A1: i1*i,i2*i are_congruent_mod i3 and
A2: i,i3 are_coprime;
  i3 divides (i1*i-i2*i) by A1;
  then i3 divides (i1-i2)*i;
  then i3 divides (i1-i2) by A2,INT_2:25;
  hence thesis;
end;
